Alba has set up seven goals for Messi this season in all competitions and Barca's all-time top scorer returned the favour with a curling pass from deep which his team mate slid in to poke inside the near post for the third goal after 28 minutes.
New signing Philippe Coutinho, who officially joined on Monday, watched on from the stands as Barca ran riot. The Brazil midfielder will make his debut after he recovers from a right leg muscle injury.
Level at 1-1 after the first leg, the three-time defending champions were 4-0 up by half time.
Messi quickly quashed any chance of Celta springing an upset. The Argentina worldwide then worked goalkeeper Sergio Alvarez with a shot from the edge of the area.
"Messi is the best player in history and the best in the world by a difference", Alba told reporters after the game.More news: Bremer Trust National Association Has $368000 Position in NXP Semiconductors NV (NXPI)
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Two minutes later the same combination linked up again as Messi poked home another Alba cross.
Not long after, Andres Iniesta was also withdrawn from the action, and given there was no need to chase more goals, Barcelona were happy to hold the ball and conserve energy for much of the second period.
Celta drew 2-2 in the Camp Nou in the La Liga earlier in the season and held Real Madrid to a 2-2 draw on Sunday.
Sevilla beat Cadiz 2-1 with goals from Wissam Ben Yedder and Joaquin Correa to advance 4-1 on aggregate.
"It was a complicated match". "But I shouldn't say that because it will mean we'll draw them!"